Hi Bruno! On 11 Feb., 01:37, Bruno Le Floch <blfla...@gmail.com> wrote: > > You could have both consistencies. That depends on how you define gcd > > and lcm: > > > - Quotient fields as described by Bruno. > > - Fields: zero if both elements are zero. A non-zero element > > otherwise (most fields would choose 1 here). > > - PID: a generator of the corresponding ideal. > > I don't see how this brings in both consistencies. Algebraic > consistency requires gcd and lcm on QQ to have different outputs > depending on whether QQ is seen a Field, a PID, a Quotient Field... Is > there a clear way for the user to indicate "which QQ" he wants?
I am sorry for the FUD that I spread in my earlier posts. Meanwhile people convinced me that it is indeed possible to have both consistencies. The point is: In a PID, "the" lcm of a and b is only defined up to a unit - it is only required that lcm(a,b) generates the intersection of the ideals generated by a and b. My mistake was: 1 is certainly "the" canonical choice of a generator of the ideal <a>\cap <b> (a,b non-zero); but that does not mean that it is the best return value of lcm(a,b)! So, if lcm(a,b) for a,b non-zero returns *any* non-zero element, then "consistency from the category point of view" is granted - lcm(1/2,1/4) = 42 is not wrong in QQ. But that freedom means: We can *in addition* achieve consistency with respect to sub-structures, namely by seeing QQ as a quotient field. Cheers, Simon -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
